Recently, Lie groupoids equipped with geometric structures suitably compatible with Morita equivalence have been object of intense research. This is due to the fact that such structures descend to the quotient stack of a Lie groupoid. For instance, in [1]} the notion of Morita equivalence of VB-groupoids plays a role to define vector bundles over differentiable stacks; Riemannian metrics on Lie groupoids were introduced in [2]} showing that they behave well with respect to Morita equivalence, hence inducing a Riemannian metric on the associated quotient stack [3]}; Lie algebroids over stacks are modeled by LA-groupoids [4]}, the space of multiplicative sections of an LA-groupoid [5]} has the structure of a Lie 2-algebra which is Morita invariant and, in particular, the space of vector fields on a differentiable stack has a natural structure of Lie 2-algebra [6]}, [5]} as conjectured in [8]}. Also, invariants of differentiable stacks can be defined in terms of their groupoid counterparts. As an example, the equivariant cohomology of a Lie group acting on a differentiable stack was introduced in [9]} by looking at the Bott-Shulmann cohomology of a certain action groupoid encoded by an equivariant atlas of a \(G\) -stack. Another invariant of a differentiable stack is introduced in [10]} by mean of the cohomology of a Lie groupoid with coefficients in a representation up to homotopy of Lie groupoids.
